Canonical quantization of 1+1 dimensional gravity

Based on Polyakov's evaluation of the Fadeev-Popov determinant for (1+1)-dimensional gravity in the conformal gauge we formulate a canonical quantization in the synchronous gauge. We find that the system is describable as a quantum mechanical system of one degree of freedom. The quantization can be carried out and solved when any number of gauge fields are included. Scalar and spinor fields lead to new difficulties. For positive cosmological constant the geometry collapses as suggested by the classical system. For negative cosmological constant a more interesting behavior involving exponentially expanding and contracting universes occurs.Supported in part by NSF Grant No. PHy 78-26847.On leave from Tel Aviv University. Supported in part by the Israel Commission for Basic Research.